U.S. patent number 10,368,738 [Application Number 14/298,176] was granted by the patent office on 2019-08-06 for fast absolute-reflectance method for the determination of tear film lipid layer thickness.
This patent grant is currently assigned to Johnson & Johnson Surgical Vision, Inc.. The grantee listed for this patent is Johnson & Johnson Surgical Vision, Inc.. Invention is credited to Stan Huth, Denise Tran.
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United States Patent |
10,368,738 |
Huth , et al. |
August 6, 2019 |
Fast absolute-reflectance method for the determination of tear film
lipid layer thickness
Abstract
A method of determining tear film lipid layer thickness. The
method includes the steps of measuring a tear film aqueous plus
lipid layer relative reflectance spectrum using a
wavelength-dependent optical interferometer; converting the
measured tear film aqueous plus lipid layer relative reflectance
spectrum to a calculated absolute reflectance spectrum; and
comparing the calculated absolute reflectance spectrum to a
theoretical absolute lipid reflectance spectrum to determine a tear
film lipid layer thickness.
Inventors: |
Huth; Stan (Newport Beach,
CA), Tran; Denise (Irvine, CA) |
Applicant: |
Name |
City |
State |
Country |
Type |
Johnson & Johnson Surgical Vision, Inc. |
Santa Ana |
CA |
US |
|
|
Assignee: |
Johnson & Johnson Surgical
Vision, Inc. (Santa Ana, CA)
|
Family
ID: |
53276279 |
Appl.
No.: |
14/298,176 |
Filed: |
June 6, 2014 |
Prior Publication Data
|
|
|
|
Document
Identifier |
Publication Date |
|
US 20150351627 A1 |
Dec 10, 2015 |
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
A61B
3/101 (20130101) |
Current International
Class: |
A61B
3/10 (20060101) |
Field of
Search: |
;351/200,205,206,207,208,209,210,215,221 |
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|
Primary Examiner: Tallman; Robert E.
Attorney, Agent or Firm: Withrow & Terranova,
P.L.L.C.
Claims
What is claimed is:
1. A method of determining tear film lipid layer thickness,
comprising the steps of: measuring a tear film aqueous plus lipid
layer relative reflectance spectrum using a wavelength-dependent
optical interferometer; converting the measured tear film aqueous
plus lipid layer relative reflectance spectrum to a calculated
absolute reflectance lipid spectrum, wherein the calculated
absolute reflectance lipid spectrum is calculated based on
wavelength-dependent refractive indices for air, lipid, and aqueous
phases and wherein the wavelength-dependent refractive indices for
lipid are based upon a Sellmeier equation form; and comparing the
calculated absolute reflectance lipid spectrum to a theoretical
absolute reflectance lipid spectrum to determine a tear film lipid
layer thickness.
2. The method of claim 1, wherein the tear film aqueous plus lipid
layer relative reflectance spectrum is measured relative to a
reference material.
3. The method of claim 2, wherein converting the measured tear film
aqueous plus lipid layer relative reflectance spectrum to the
calculated absolute reflectance lipid spectrum comprises
multiplying the measured tear film aqueous plus lipid layer
relative reflectance spectrum at a plurality of wavelengths times
an absolute reference lipid spectrum for the reference material
obtained at the plurality of wavelengths.
4. The method of claim 3, wherein the reference material has a
radius of curvature of 7.75 mm.
5. The method of claim 3, wherein converting the measured tear film
aqueous plus lipid layer relative reflectance spectrum to the
calculated absolute reflectance lipid spectrum further comprises
dividing by a correction factor to correct for differences between
the theoretical absolute reflectance lipid spectrum and the
calculated absolute reflectance lipid spectrum.
6. The method of claim 3, wherein converting the measured tear film
aqueous plus lipid layer relative reflectance spectrum to the
calculated absolute reflectance lipid spectrum further comprises
dividing by 100.
7. The method of claim 1, wherein comparing the calculated absolute
reflectance lipid spectrum to the theoretical absolute reflectance
lipid spectrum comprises iteratively comparing the calculated
absolute reflectance lipid spectrum to a plurality of theoretical
absolute reflectance lipid spectra.
8. The method of claim 7, wherein iteratively comparing the
calculated absolute reflectance lipid spectrum to the plurality of
theoretical absolute reflectance lipid spectra comprises minimizing
a sum of least squares differences between the calculated absolute
reflectance lipid spectrum and the plurality of theoretical
absolute reflectance lipid spectra.
9. The method of claim 8, wherein minimizing the sum of least
squares differences comprises using a Levenburg-Marquardt
algorithm.
10. The method of claim 7, wherein iteratively comparing the
calculated absolute reflectance lipid spectrum to the plurality of
theoretical absolute reflectance lipid spectra comprises starting
the iteration with the theoretical absolute reflectance lipid
spectrum for a 65 nm thick lipid layer.
11. The method of claim 1, wherein comparing the calculated
absolute reflectance lipid spectrum to the theoretical absolute
reflectance lipid spectrum does not include a comparison of the
calculated absolute reflectance lipid spectrum to a theoretical
absolute reflectance aqueous spectrum to determine a tear film
aqueous layer thickness and a comparison of the calculated absolute
reflectance lipid spectrum to a theoretical film stack to determine
corneal surface refractive index.
12. A system for determining tear film lipid layer thickness,
comprising: a wavelength-dependent optical interferometer; and a
controller in communication with the wavelength-dependent optical
interferometer, the controller configured to measure a tear film
aqueous plus lipid layer relative reflectance spectrum using the
wavelength-dependent optical interferometer, convert the measured
tear film aqueous plus lipid layer relative reflectance spectrum to
a calculated absolute reflectance lipid spectrum, wherein the
calculated absolute reflectance lipid spectrum is calculated based
on wavelength-dependent refractive indices for air, lipid, and
aqueous phases and wherein the wavelength-dependent refractive
indices for lipid are based upon a Sellmeier equation form, and
compare the calculated absolute reflectance lipid spectrum to a
theoretical absolute reflectance lipid spectrum to determine a tear
film lipid layer thickness.
13. The system of claim 12, wherein the tear film aqueous plus
lipid layer relative reflectance spectrum is measured relative to a
reference material.
14. The system of claim 13, wherein the controller, in order to
convert the measured tear film aqueous plus lipid layer relative
reflectance spectrum to a calculated absolute reflectance lipid
spectrum, is further configured to multiply the measured tear film
aqueous plus lipid layer relative reflectance spectrum at a
plurality of wavelengths times an absolute reference lipid spectrum
for the reference material obtained at the plurality of
wavelengths.
15. The system of claim 14, wherein the reference material has a
radius of curvature of 7.75 mm.
16. The system of claim 14, wherein the controller, in order to
convert the measured tear film aqueous plus lipid layer relative
reflectance spectrum to the calculated absolute reflectance lipid
spectrum, is further configured to divide by a correction factor to
correct for differences between the theoretical absolute
reflectance lipid spectrum and the calculated absolute reflectance
lipid spectrum.
17. The system of claim 14, wherein the controller, in order to
convert the measured tear film aqueous plus lipid layer relative
reflectance spectrum to the calculated absolute reflectance lipid
spectrum, is further configured to divide by 100.
18. The system of claim 12, wherein the controller, in order to
compare the calculated absolute reflectance lipid spectrum to the
theoretical absolute reflectance lipid spectrum, is further
configured to iteratively compare the calculated absolute
reflectance lipid spectrum to a plurality of theoretical absolute
reflectance lipid spectra.
19. The system of claim 18, wherein the controller, in order to
iteratively compare the calculated absolute reflectance lipid
spectrum to the plurality of theoretical absolute reflectance lipid
spectra, is further configured to minimize a sum of least squares
differences between the calculated absolute reflectance lipid
spectrum and the plurality of theoretical absolute reflectance
lipid spectra.
20. The system of claim 19, wherein the controller, in order to
minimize the sum of least squares differences, is further
configured to use a Levenburg-Marquardt algorithm.
21. The system of claim 18, wherein the controller, in order to
iteratively compare the calculated absolute reflectance lipid
spectrum to the plurality of theoretical absolute reflectance lipid
spectra, is further configured to start the iteration with the
theoretical absolute reflectance lipid spectrum for a 65 nm thick
lipid layer.
22. The system of claim 12, wherein the controller, in order to
compare the calculated absolute reflectance lipid spectrum to the
theoretical absolute reflectance lipid spectrum, is further
configured not to include a comparison of the calculated absolute
reflectance lipid spectrum to a theoretical absolute reflectance
aqueous spectrum to determine a tear film aqueous layer thickness
and a comparison of the calculated absolute reflectance lipid
spectrum to a theoretical film stack to determine corneal surface
refractive index.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
The present application is related to U.S. patent application Ser.
No. 14/298,036 to Huth and Tran, "Method for Rapid Calculation of
Tear Film Lipid and Aqueous Layer Thickness and Ocular Surface
Refractive Index from Interferometry Spectra," filed Jun. 6, 2014,
now issued as U.S. Pat. No. 9,456,741 which is incorporated herein
by reference in its entirety.
BACKGROUND
The present invention relates to determination of tear film lipid
layer thickness.
Dry eye disease is most often caused by excessive tear film
evaporation, leading to hyperosmolarity of the tear film, resulting
in ocular surface inflammation and exacerbation of the problem.
Excessive tear film evaporation is often caused by an abnormal tear
film lipid layer, either in amount or in quality. The amount or
quality of tear film lipid can manifest itself in changes in
thickness of the lipid layer. Generally, a thicker lipid layer is
associated with a normal tear film, whereas the opposite is often
the case for dry eye. Present clinical measurements of the tear
film lipid layer are for the most part qualitative or
semi-quantitative in nature. Korb, in U.S. Pat. Nos. 8,591,033 and
8,585,204, disclose a quantitative method for measuring the
thickness of the tear film lipid layer. However, this method does
not measure the lipid layer over the central cornea where tear film
thinning and breakup due to evaporation is maximal and where it is
believed a better diagnosis of dry eye can be obtained. Huth, in
U.S. Pat. No. 8,602,557 B2 (incorporated herein by reference in its
entirety) also disclose a quantitative method for measuring the
thickness of the tear film lipid layer as part of a method to
simultaneously measure the tear film aqueous layer and the corneal
surface refractive index. However, this method requires as long as
475 seconds to complete the calculations for a single tear film
spectrum.
SUMMARY
Thus, it is the object of the present invention to overcome the
limitations of the prior art, and to increase the sensitivity,
accuracy and precision of the measurement of the tear film lipid
layer. Fast, accurate and precise lipid layer
thickness-determination methods are also needed for the
quantitative evaluation of the effects of novel dual-function
lipid-supplementation tear formulas on the tear film lipid layer.
Such methods are also needed to evaluate the effects of other eye
drops, ophthalmic dry eye drugs and MPS solutions, and contact
lenses on the tear film lipid layer.
In one embodiment, the invention provides a method of determining
tear film lipid layer thickness. The method includes the steps of
measuring a tear film aqueous plus lipid layer relative reflectance
spectrum using a wavelength-dependent optical interferometer;
converting the measured tear film aqueous plus lipid layer relative
reflectance spectrum to a calculated absolute reflectance spectrum;
and comparing the calculated absolute reflectance spectrum to a
theoretical absolute lipid reflectance spectrum to determine a tear
film lipid layer thickness.
In another embodiment the invention provides a system for
determining tear film lipid layer thickness. The system includes a
wavelength-dependent optical interferometer and a controller. The
controller is in communication with the interferometer and is
configured to measure a tear film aqueous plus lipid layer relative
reflectance spectrum using the interferometer, convert the measured
tear film aqueous plus lipid layer relative reflectance spectrum to
a calculated absolute reflectance spectrum, and compare the
calculated absolute reflectance spectrum to a theoretical absolute
lipid reflectance spectrum to determine a tear film lipid layer
thickness.
Other aspects of the invention will become apparent by
consideration of the detailed description and accompanying
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows absolute reflectance for a 200 nm tear lipid film
layer measured in a wavelength range of 575 nm to 1075 nm.
FIG. 2 shows calculated absolute reflectance spectra for lipid
layers thicknesses in a range of 0-140 nm.
FIG. 3 shows an unmodified tear film lipid spectrum obtained using
an interferometer.
FIG. 4 shows absolute reflectance spectra for SiO.sub.2 layers
having thicknesses from 0-188.6 nm.
FIG. 5 shows the results of converting measured relative SiO.sub.2
spectra to calculated absolute reflectance spectra, where each
absolute reflectance spectrum obtained from measured data is
compared to an equivalent calculated theoretical spectrum.
FIG. 6 shows the comparison between a calculated theoretical
spectrum for a 48.26 nm SiO2 thin film standard compared to
calculated absolute reflectance spectra obtained by converting
measurements of the relative reflectance spectra for the 48.26 nm
SiO.sub.2 thin film standard which were measured relative to
various reference materials, including a flat silicon wafer, a flat
BK7 glass plate, and a curved BK7 glass lens having an identical
radius of curvature as the human cornea, i.e. 7.75 mm.
FIG. 7 shows a standard curve comparing the actual thickness (in
nm) of the SiO2 thin film standards to the thickness determined
using the methods disclosed herein.
FIG. 8 shows a theoretical spectrum (dashed line, ----) vs. b-term
fitted spectrum (light grey line, overlapping dashed line) and
spectrum without the b-term fit (black line).
FIGS. 9 and 10 show calculated absolute reflectance spectra for the
MgF.sub.2 coating on the convex (FIG. 9) and the flat (FIG. 10)
faces of a coated BK7 lens, both compared to a calculated
theoretical spectrum for the MgF.sub.2 coating.
FIG. 11 shows calculated absolute reflectance spectra for tear film
lipid spectra compared to calculated theoretical spectra.
FIG. 12 shows thicknesses of the lipid (diamonds) vs. aqueous
(squares) tear film layers during blinking (downward spikes in
lipid thickness measurements) of the subject's eyelids.
DETAILED DESCRIPTION
Before any embodiments of the invention are explained in detail, it
is to be understood that the invention is not limited in its
application to the details of construction and the arrangement of
components set forth in the following description or illustrated in
the following drawings. The invention is capable of other
embodiments and of being practiced or of being carried out in
various ways.
Human tear film lipid layer thickness is believed to be between
20-200 nm. Central corneal tear film lipid layer thickness rarely
exceeds about 120 nm, however, and also can be less than 20 nm in
thickness. Normal wavelength-dependent optical interferometric
methods for the determination of thin film thickness, based upon
the analysis of the increasing number of cosine-function spectral
oscillations with thickness, are unsuitable for this range. This is
so because even at 200 nm thickness, only half an oscillation is
visible within the 575-1075 nm spectral wavelength range of the
typical optical interferometer (FIG. 1).
Accordingly, the answer to this problem is to base the thicknesses
upon absolute optical reflectivity, also illustrated in FIG. 1,
where absolute reflectivity of a 200 nm lipid layer is 0.02=2% at
575 nm. Thus, one needs to compare the absolute reflectivity
spectrum of a tear lipid layer to a standard reflectivity spectrum
such as seen in FIG. 1. The approach to achieving this is disclosed
herein.
Interferometry measurements are typically determined as relative
percent light reflection values, where the measurements from a
subject's tear film and cornea are expressed relative to
measurements obtained from a reference material (e.g. a particular
glass material having a radius of curvature comparable to that of a
cornea, e.g. ranging from 7-9 mm, in particular 7.75 mm). The ratio
of the light reflectance from the subject's tear film and cornea,
R(.lamda.) sample, to the light reflectance from the reference,
R(.lamda.) reference, is multiplied by 100 by typical spectrometer
and CCD detector software, so that the final relative light
reflection values are expressed as percentages; the percentage
values are determined for a range of wavelengths to obtain a
spectrum. Thus, the y-axis of a measured spectrum corresponds to
100.times.R(.lamda.) sample/R(.lamda.) reference.
As an initial step in the development of the methods of the present
invention, the theoretical absolute spectra for lipid layers of
various thicknesses were calculated. Given that these theoretical
absolute spectra have similar shapes or slopes to one another, but
different absolute optical reflectivity at different wavelengths,
correlating the theoretical absolute spectra with observed spectra
requires determining the absolute reflectance spectrum from the
measured data and correlating the absolute spectra with the
theoretical spectra. Experimentation is required, however, to
account for discrepancies between theoretical predicted spectra and
actual measured spectra. Lipid layer standards of known thicknesses
are not available and so it is not possible to perform calibrations
using lipids. Instead, a series of calibrations was performed using
commercially-available standards having layers of silicon dioxide
of known thicknesses, comparable to thicknesses of tear film lipid
layers. The measured spectra for the silicon dioxide standards were
compared to the predicted theoretical absolute spectra for silicon
dioxide layers of the same thicknesses and a correction algorithm
was produced which can be used to obtain the absolute reflectance
values for measured tear film lipid layers. The correction
algorithm accounts for changes in light reflection arising from the
geometry of a curved reference lens or from non-orthogonal
placement of a flat reference surface with respect to incident
light and from out-of-focus light reflection.
In various embodiments, the disclosed methods are modifications of
a procedure for calculation of absolute reflectance of the presumed
tear structure, involving an air interface with a single lipid
layer overlying an aqueous layer.
In the aforementioned procedure, n.sub.0, n.sub.1, and n.sub.2 are
the refractive indices of the air, lipid layer, and aqueous layer,
respectively, for which fixed values of n.sub.0=1, n.sub.1=1.48,
and n.sub.2=1.33 have been used. In one embodiment of the present
invention, the respective complex refractive indices are used for
n.sub.1 and n.sub.2, each of which changes with wavelength.
The Fresnel indices of reflection r.sub.1 and r.sub.2 for the
air-lipid and lipid-aqueous interfaces are, respectively:
r.sub.1=(n.sub.0-n.sub.1)/(n.sub.0+n.sub.1) and
r.sub.2=(n.sub.1-n.sub.2)/(n.sub.1+n.sub.2)
Since energy is proportional to the square of amplitude,
R(.lamda.)=R.times.R*=|R.sup.2|, where R* is the conjugate complex
numbers of R. Thus, from Euler's equation:
R(.lamda.)=(r.sub.1.sup.2+r.sub.2.sup.2+2r.sub.1r.sub.2 cos
2.delta..sub.1)/(1+r.sub.1.sup.2r.sub.2.sup.2+2r.sub.1r.sub.2 cos
2.delta..sub.1)=
1-(8n.sub.0n.sub.1.sup.2n.sub.2)/((n.sub.0.sup.2+n.sub.1.sup.2)(n.sub.1.s-
up.2+n.sub.2.sup.2)+4n.sub.0n.sub.1.sup.2n.sub.2+(n.sub.0.sup.2-n.sub.1.su-
p.2)(n.sub.1.sup.2-n.sub.2.sup.2)cos 2.delta..sub.1)
where a phase difference between two waves r.sub.1 and r.sub.2, is
2.delta..sub.1 and 2.delta..sub.1=(4.pi./.lamda.)n.sub.1d cos
.phi..sub.1
where .phi..sub.1 is the angle of refraction of the incident light
upon the lipid layer,
which=9.369.degree. for the wavelength-dependent optical
interferometer used in the methods of the present invention, thus
cos .phi..sub.1=0.986659.
The complex refractive indices for n.sub.1 (lipid) and n.sub.2
(aqueous) are used: n.sub.1=
nd=y=sqrt(1+(((-851.03)*x*x)/(x*x-(816.139)))+(((420.267)*x*x)/(x*x-(-706-
.86))) plus(((431.856)*x*x)/(x*x-(2355.29))))
where x=wavelength
and where nd=Sellmeier equation form of tear meibomian lipid
refractive index derived from primary refractive index data from
Tiffany, J M. Refractive index of meibomian and other lipids.
Current Eye Research. 5(11), 1986; 887-889: 430 nm: nd=1.5126; 450
nm:1.5044; 510 nm: 1.4894; 590 nm:1.4769; 710 nm:1.4658. The
Sellmeier equation coefficients were derived by fitting the limited
Tiffany data (no data exist beyond 710 nm, where most of the
spectral range of the interferometer exists (spectral range:
559-1085 nm)) first to a polynomial
(nd=5.04e-7.times.2-0.00736x+1.73481, wherein x=wavelength) to
generate forecasted refractive index values at 600, 620, 640 and
680 nm, followed by fitting the Tiffany+forecasted refractive index
data set to the Sellmeier equation format wherein refractive index
data beyond 710 nm could be calculated and utilized. The Sellmeier
equation format for lipid refractive index data is believed to
provide more accurate refractive index information, which is
critical for accurate lipid layer thickness calculations.
n.sub.2=1.32806+0.00306*(1000/.lamda.)^2
Using the above equations, an Excel spreadsheet is created in which
the values of the three refractive indices n.sub.0, n.sub.1 and
n.sub.2 are calculated (except for n.sub.0, which is always 1) for
the wavelengths within the wavelength range measured by the
interferometer (e.g., for wavelengths between 559-1085 nm). Then,
using the expanded Euler's equation with all terms, R(.lamda.) is
calculated for a series of lipid layer thicknesses, d. The results
from 575-1075 nm are seen in FIG. 2, presenting a series of
absolute-value reflectance spectra for lipid layers of various
thicknesses.
The spectrum (not shown) for a lipid layer of 0 nm thickness is
very flat, lies just below that of a 20 nm thick lipid layer and
produces an absolute reflectance of 0.02% of the incident light at
550 nm. It can be seen in FIG. 2 that there is little difference in
the slopes of the lipid layer spectra between 20-100 nm
thicknesses, further reinforcing the need to base lipid layer
thickness calculations upon absolute reflectance values rather than
parameters such as the slope or shape of the spectra. Thus, a 20 nm
lipid layer will reflect about 0.025=2.5% of the incident light at
550 nm, whereas a 100 nm lipid layer will reflect about 0.06=6% of
the incident light at 550 nm. At 120 nm thickness and larger, the
slope of absolute reflectance begins to change from a negative
slope to a flat slope. Beyond 140 nm thickness, as seen in FIG. 1
for a 200 nm thickness, the slope becomes positive. Slope
evaluation at 120 nm and beyond becomes a valuable tool to
distinguish between a thin or thick lipid layer.
Thus, for the majority of tear film lipid layers between 0-100 nm
thickness, it is not a simple exercise to measure a tear lipid
spectrum with a wavelength-dependent optical interferometer and
compare it to one of the above absolute reflectance spectra. This
is illustrated in FIG. 3, which shows an unmodified measured tear
lipid spectrum.
Accordingly, each measured spectrum must be converted to a
calculated absolute spectrum and then compared to the absolute
reflectivity derived from theory. This comparison must be
accomplished mathematically.
This new procedure required development and validation with thin
film standards. Since thin lipid film standards are not available,
thin SiO.sub.2 film standards, produced via vapor deposition of
SiO.sub.2 onto pure flat Silicon wafer substrates were used. These
standards are commercially available (VLSI Standards, Inc. San
Jose, Calif. 95134-2006) and calibrated to within 0.1-0.01 nm
thickness by NIST. The following SiO.sub.2 standards were employed
(the 0 nm standard was a pure silicon wafer without SiO.sub.2;
Table 1).
TABLE-US-00001 TABLE 1 SiO.sub.2 Actual thickness, nm 0 48.26 95.01
188.58
Absolute reflectivity of these SiO.sub.2 films were calculated,
using the same procedure with the expanded Euler equation above,
substituting the complex refractive indices below for n.sub.1 and
n.sub.2 (n.sub.0=air=1, as before).
SiO.sub.2n.sub.1=-9.3683E-11x.sup.3+2.5230E-07x.sup.2-2.3810E-04x+1.5302E-
+00
Sin.sub.2=SQRT(1+((5.66474*.lamda.*.lamda.)/((.lamda.*.lamda.)-119153)-
)+((5.29869*.lamda.*.lamda.)/(.lamda.*.lamda.-(51556.1)))+((-24642*.lamda.-
*.lamda.)/((.lamda.*.lamda.)-(-146300000000))))(conversion of raw
n.sub.2 data to the Sellmeier equation form)
The latter equation for the refractive index of pure silicon is in
the form of the Sellmeier equation. This equation form is
considered to provide very accurate values of the refractive index
as a function of wavelength. Not all refractive index data are
provided in this form, however. The resulting absolute reflectance
spectra are seen in FIG. 4.
Interferometer-measured spectra for SiO.sub.2 standards, as for
measured tear film lipid spectra, are expressed as relative %
reflectance, since the measured reflectance from a thin film is
measured relative to the measured reflectance from a reference. The
y-axis of a measured spectrum corresponds to 100.times.R(.lamda.)
meas. sample/R(.lamda.) meas. reference.
A number of different references can be used, although the best
reference for tear film spectra is one with the same radius of
curvature as the cornea (r=7.75 mm), to allow for the same
reflectance geometry. In certain embodiments, a spherically-curved
BK7 glass reference lens is used for this purpose. A pure flat
silicon wafer can also be used as the reference for the SiO.sub.2
standards, since both surfaces are flat. The conversion procedure
to convert a measured SiO.sub.2 or tear film spectrum to a
calculated absolute reflectance spectrum is to first divide by 100
and then to multiply the (R(.lamda.) meas. Sample/R(.lamda.) meas.
Reference).times.R(.lamda.) absolute reference. Here, R(.lamda.)
absolute reference is the calculated theoretical reflectivity of
the reference and the abbreviation for absolute when used
throughout this disclosure will be: abs. This result can then be
compared mathematically to a theoretical SiO2 or tear lipid
spectrum, as illustrated in FIG. 4 for SiO.sub.2. In order to
accomplish this procedure, one has to first calculate R(.lamda.)
absolute reference (for pure Silicon or BK7). The equation which is
used is derived from the following equations.
If incident light is unpolarized, and since R=|r|.sup.2 (since
reflected intensity is proportional to the square of the modulus of
the electric field amplitude and the dielectric function), then
total R=(r.sub.s.sup.2+r.sub.p.sup.2)/2.
Also, from Snell's law, where sin .psi.=(n.sub.1/n.sub.2) sim
.phi., cos
.psi.=(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup.1/2.
Then, from the theory of interface reflection between two isotropic
materials (e.g. air and an isotropic solid such as Si or BK7), the
indices of reflection are determined as follows: r.sub.s=(n.sub.1
cos .phi.-n.sub.2 cos .psi.)/(n.sub.1 cos .phi.+n.sub.2 cos .psi.)=
(n.sub.1 cos
.phi.-n.sub.2(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup.1/2)-
/(n.sub.1 cos
.phi.+n.sub.2(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup.1/2),
and r.sub.p=(n.sub.2 cos .phi.-n.sub.1 cos .psi.)/(n.sub.2 cos
.phi.+n.sub.1 cos .psi.)= (n.sub.2 cos
.phi.-n.sub.1(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup.1/2)/(n.-
sub.2 cos
.phi.+n.sub.1(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup-
.1/2)
These equations may also be written as: r.sub.s=(n.sub.1 cos
.phi.-(n.sub.2.sup.2/n.sub.1.sup.2
sin.sup.2.phi.).sup.1/2)/(n.sub.1 cos
.phi.+(n.sub.2.sup.2-n.sub.1.sup.2 sin.sup.2.phi.).sup.1/2),
since(n.sub.2.sup.2-n.sub.1.sup.2
sin.sup.2.phi.).sup.1/2)=n.sub.2((n.sub.2.sup.2/n.sub.2.sup.2)-(n.sub.1.s-
up.2/n.sub.2.sup.2)sin.sup.2.phi.).sup.1/2=n.sub.2(1-(n.sub.1.sup.2/n.sub.-
2.sup.2)sin.sup.2.phi.).sup.1/2 and r.sub.p=(n.sub.2 cos
.phi.-n.sub.1(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup.1/2)/(n.-
sub.2 cos
.phi.+n.sub.1(1-(n.sub.1.sup.2/n.sub.2.sup.2)sin.sup.2.phi.).sup-
.1/2)
Then, since n.sub.1=air=1, the equations above can be combined
using the relationship R
abs=(r.sub.s.sup.2+r.sub.p.sup.2)/2=R(.lamda.) abs Si or R(.lamda.)
abs BK7 (i.e. to obtain calculated absolute reflectance values for
silicon or BK7 glass): R(.lamda.)abs
Si=(((0.986659-D3*SQRT(1-(0.162799/D3)^2))/(0.986659+D3*SQRT(1-(0.162799/-
D3)^2)))^2+(((SQRT(1-(0.162799/D3)^2))-D3*0.986659)/((SQRT(1-(0.162799/D3)-
^2))+D3*0.986659))^2)/2
where D3=Sellmeier refractive index for Si at each .lamda.,
and where 0.986659=cos .phi. and
0.162799=sin .phi. and where each wavelength is an exact wavelength
measured by the interferometer.
R(.lamda.) abs BK7 is calculated for each wavelength using the same
equation, except D3=Sellmeier refractive index for BK7 at each
.lamda.. Sellmeier
BK7n=SQRT(1+(1.03961212*G3*G3)/(G3*G3-0.00600069867)+(0.2317923-
44*G3*G3)/(G3*G3-0.0200179144)+(1.01046945*G3*G3)/(G3*G3-103.560653))
where G3=interferometer wavelength in microns (BK7 refractive index
ref=Schott technical information document TIE-29 (2005)).
FIG. 5 shows the results of converting measured SiO.sub.2 spectra
(using a 4V or 4.5V light source voltage, 20 msec single scan) to
calculated absolute reflectance spectra for various silicon
reference standards. The measured relative spectra are first
divided by 100 and then multiplied times R(.lamda.) abs Si
reference (Si abs R calculated) to obtain R(.lamda.) abs SiO.sub.2
sample spectra.
It can be seen that there are some relatively small differences
between the theoretical and calculated absolute spectra. These
differences can be mathematically calculated using an algorithm
which compares the calculated absolute spectra to theoretical
absolute spectra of various SiO.sub.2 films of varying thicknesses.
In one particular embodiment, this is accomplished by creating a
Statistica software program (StatSoft.RTM., Tulsa, Okla.) based
upon the expanded Euler equation from above:
V5=(1-48*v1*v2**2*v3)/((v1**2+v2**2)*(v2**2+v3**2)+4*v1*v2**2*v3+(-
(v1**2-v2**2)*(v2**2-v3**2)*(cos(4*3.14159*v2*a*0.98666/v4))))))
where
v5=R(.lamda.) SiO.sub.2 measured sample.times.R(.lamda.) abs Si
reference (Si abs R calculated)/100
and where
v1=n.sub.0 air=1,
v2=n.sub.1(.lamda.) SiO.sub.2,
v3=n.sub.2 (.lamda.) Si,
v4=measured .lamda., and
the variable a, the fitted film thickness.
As shown below, the wavelength range for SiO.sub.2 thin film
standards must be limited to 575-1025 nm, to avoid signal
weakness/potential optical aberration at the wavelength extremes.
Thus, the program requires the input of five columns of calculated
and measured data input as variables (i.e. v1-v5). The Non-linear
Estimation method within the Statistica software is used, wherein
the equation for v5 above is input as the function to be estimated
into the space provided in the user specified regression, least
squares module. The Statistica software program uses the
Levenburg-Marquardt algorithm to achieve a minimum in the sum of
squares of the differences between theoretical R and the product of
R(.lamda.) SiO.sub.2 measured sample.times.R(.lamda.) abs Si
reference (Si abs R calculated)/100 at each wavelength. In various
embodiments, other mathematical algorithms for fitting data are
available within Statistica and other software platforms and can
also be employed. This software module requires the number of
calculation/fitting iterations to be selected. In one embodiment,
fifty iterations were found to be acceptable, although other lower
or higher numbers of iterations are also acceptable and can be
readily determined by an evaluation of the p-level of the fit. All
p-levels for thin films were found to be 0.00 and are thus highly
significant.
The program also requires a starting value for the variable a-term,
the fitted film thickness. This is indicated in the column labeled
"STAT Input" in Table 2 below for the SiO.sub.2 thin film
standards. It was discovered through experiments with tear film
spectra, presented below, that a starting value for the a-term that
is too far from the actual thin film thickness value may converge
to an incorrect result. It is believed that this may be a result of
convergence to a local minimum in the least squares sum.
Considerable time may be required to run the program multiple times
with different starting values until the correct value is found.
Moreover, the correct tear film lipid layer thickness is not known
prior to calculation and thus the correct starting value is not
known. Thus, selecting the proper starting value is important not
only for a fast method, but also to achieve correct results. The
human tear film lipid layer thickness typically ranges between 0
and 120 nm in thickness. Thus, in one particular embodiment a
starting value of 65 nm has been determined to produce correct
results for tear film lipid layer thickness between 0 and about 100
nm thickness as well as for SiO.sub.2 thin film standards between 0
and 95 nm. Other starting values may be employed where necessary in
cases where a 65 nm starting value is incorrect, as is the case for
the 188.58 nm SiO.sub.2 thin film standard, in which case a 200 nm
starting value was used.
Subsequently, the validity of the method was demonstrated using a
curved BK7 glass reference lens having an identical radius of
curvature as the human cornea, 7.75 mm. This particular lens is
used as a reference when making measurements of the human tear film
in order to achieve optics as close as possible to those during the
human tear film measurements.
A similar mathematical process was completed using measured
SiO.sub.2 thin film standard spectra with different reference
materials. FIG. 6 shows absolute reflectance spectra obtained by
converting measurements of the relative reflectance spectra for the
48.26 nm SiO.sub.2 thin film standard which were measured relative
to various reference materials, including a flat silicon wafer, a
flat BK7 glass plate, and a curved BK7 glass lens having an
identical radius of curvature as the human cornea, i.e. 7.75 mm.
The absolute spectra, determined from measured relative spectra,
are graphed relative to the calculated absolute spectrum for a
48.26 nm SiO.sub.2 thin film. FIG. 6 shows that when either a flat
Silicon wafer or flat BK7 reference lens is used instead of the
curved BK7 lens, the spectra determined from the measured relative
reflectance spectra closely match the respective theoretical
spectrum. However, when the curved BK7 glass lens was used as the
reference, the absolute reflectance spectrum that was obtained does
not overlay the theoretical spectrum and instead is shifted
downward. Without being limited as to theory, this is likely
because the geometry of light reflection from the flat 48.26 nm
SiO.sub.2 and the flat BK7 standards is not the same as the light
reflection from the curved BK7 glass lens.
Similar results were obtained with the other SiO.sub.2 thin film
standards (not shown). What became evident is that a final
multiplier term (b) is required. The measured spectra must be
multiplied by the b-term to match the theoretical spectra.
Moreover, the b-term is a variable which changes between measured
spectra. The b-term is in essence a light focusing term and
corrects for non-identical focusing between the reference lens
measurement and the human tear film measurement. Thus, the expanded
Euler equation becomes:
V5=(1-((8*v1*v2**2*v3)/((v1**2+v2**2)*(v2**2+v3**2)+4*v1*v2**2*v3+((v1**2-
-v2**2)*(v2**2-v3**2)*(cos(4*3.14159*v2*a*00.98666/v4))))))*b
where
v5=R(.lamda.) SiO.sub.2 measured sample.times.R(.lamda.) abs Si
reference (Si abs R calculated)/100
and where
v1=n.sub.0 air=1,
v2=n.sub.1(.lamda.) SiO.sub.2,
v3=n.sub.2 (.lamda.) Si,
v4=measured .lamda., and
the variable a, the fitted film thickness, and
the variable b, the final correction term which moves the measured
spectrum up or down on the theoretical R axis (y-axis) to achieve a
match with theory.
In one embodiment, the software program requires the b-term to be
on the right-hand side of the equation and at the end of the
equation, since it requires starting input values for any variables
(here a and b) to be in the same order (left to right) in which
they appear in the equation program line and the variable input
value program line. Since the right side of the above equation is
the calculated theoretical reflectance which iteratively matches
the calculated measured absolute reflectance on the left hand side,
the b-term may have a value less than 1 (a b-term on the right side
of 0.5 would be equivalent to a b-term on the left side of 2.0). It
was determined through experiments varying the starting value of
the b-term from 0 to 1.30 that the b-term value can start at a
relatively wide range of values, for example between 0.40-0.80 or
between 0 and 1.30, so that the program achieves the correct
thickness. The center of the b-term range is about 0.66 and thus
this is a good starting value.
All scan times herein for the SiO.sub.2 thin film standards are
single 20 msec scans, whereas tear film spectra are typically sums
of twelve 21 msec scans. The program is very fast, calculating
results for a single spectrum in under a second and results for 50
spectra in 11 seconds. The results for the SiO.sub.2 thin film
standards are shown in Table 2 below.
TABLE-US-00002 TABLE 2 Avg. Actual Meas. Meas. SiO2 SiO2 SiO2
Thickness, STAT Thickness, Thickness, nm Input nm nm B 0 65 a 4.42
0.66 b 1.002 48.26 65 a 47.57 48.12 0.66 b 1.026 48.26 65 a 48.37
0.66 b 0.994 48.26 65 a 48.43 0.66 b 1.035 95.01 65 a 98.54 98.56
0.66 b 1.062 95.01 65 a 98.57 0.66 b 1.062 95.01 65 a 98.55 0.66 b
1.063 188.58 200 a 190.53 190.55 0.66 b 1.033 188.58 200 a 190.55
0.66 b 1.032 188.58 200 a 190.56 0.66 b 1.030
FIG. 7 shows a standard curve comparing the actual thickness (in
nm) of the SiO.sub.2 standards to the thickness determined using
the methods disclosed herein. As can be seen, the values obtained
using the disclosed methods closely match the actual thicknesses of
the silicon dioxide standards. These are excellent results, with
only a 1.8 nm error on average, maximum error of 3.6 nm and
standard curve and slope=1.0014 and 1.9156 nm, respectively,
demonstrating that the basic mathematics of the disclosed methods
are correct. The observed absolute error results are expected to
decrease with higher scan-number values for each standard and
longer scan times.
Table 3 shows the results of experiments with tear film
interferometry spectra and demonstrates that using a starting value
for the tear film lipid layer thickness (i.e. the a-term) which is
too far from the actual lipid thickness value can produce incorrect
results. Lipid thickness values for these spectra were verified
with the '557 method. Starting values for the b-term in all cases
were 0.66. Spectrum subj18rt11 is that of a tear film during Oasys
contact lens wear, indicating that the method of the present
invention is suitable for measuring tear film lipid layers during
contact lens wear.
TABLE-US-00003 TABLE 3 Statistica Statistica lipid thickness b-term
Stat lipid starting Spectrum result, nm result thickness (a-term,
nm) sub7base#49 82.17 0.2738 50, 90, 100 ok sub1#118 25.26 0.9987
65, 75 ok; 100 no: 130.79 w b = .4416 sub2AY 51.99 0.9949 40, 50,
65, 75 ok; 100 no: 102.95 w b = .6396 sub21#43 31.78 0.3701 40, 50,
65, 70 ok; 75 no: 110.86 w b = .1828 sub2CV 78.18 0.9974 50, 65,
75, 100 ok subj18rt11 7.94 0.9322 65 ok; 73, 100 no: 132.28 w b =
.3757
FIG. 8 shows the results of using the methods disclosed herein to
measure a thin MgF.sub.2 film on a curved BK7 lens surface. This
sample serves as a surrogate for a human tear lipid film on the
curved aqueous corneal surface. The results show the MgF.sub.2 film
to be 111.98 nm thick. Here, the b-term started at 1 and converged
to a value of 1.0473. The measured and theoretical spectra overlay
one another exactly, further confirming the method herein. FIG. 8
shows a theoretical spectrum (dashed line, ----) vs. b-term fitted
spectrum (light grey line, overlapping dashed line) and spectrum
without the b-term fit (black line). Materials: MgF.sub.2-coated
BK7 lens, Edmund Optics (Barrington N.J. 08007) part 49-886, radius
of curvature=7.75 mm, used with uncoated BK7 ref lens, Edmund
Optics part 49-876, radius of curvature=7.75 mm.
Given that the above results were obtained using the novel methods
disclosed herein, there is potential uncertainty as to whether the
MgF.sub.2 film thickness is 111.98 nm. Thus, a more rigorous method
confirmation experiment was conducted using a 12.7 mm diameter,
7.75 mm radius of curvature MgF.sub.2-coated plano-convex BK7 lens
(Edmunds part no. 49-855). Using the methods disclosed herein with
an interferometer light source voltage of 5.5V, the MgF.sub.2
coating thickness was measured on both sides. The convex side was
measured using a 7.75 mm radius of curvature uncoated BK7 lens as
reference and the b-term mathematics process was employed. The
plano (flat) side was measured using a flat uncoated BK7 reference
lens. The thickness calculations for flat samples do not always
require the use of the b-term. The b-term is useful when employing
flat references when the reference surface is not placed
orthogonally to the incident light from the interferometer.
Otherwise, the mathematics are identical to those used for curved
surfaces. FIGS. 9 and 10 present the results. In FIG. 9, Dark
triangles=% R/100.times.BK7 R; Light lines=Fitted R/(b=1.0585); and
Dark x=Theory R. In FIG. 10, Light triangles=% R/100.times.BK7 R;
and Dark x=Theory R.
The methods herein determined MgF.sub.2 coating thicknesses to be
110.49 nm and 108.73 nm for the convex and flat sides,
respectively. This is consistent with the assumption that the
convex and flat surfaces were coated identically. As a further
confirmation of the thickness of the MgF.sub.2 coating, a
spectroscopic Ellipsometer (model alpha SE, J. A. Woollam, Lincoln,
Nebr. 68508-2243) was employed to measure thickness of the coating
on the flat surface. However, the convex surface coating could not
be measured with the ellipsometer due to beam geometry requiring
flat samples. The ellipsometer measured a coating thickness for the
flat surface of 110.0.+-.0.64 nm, in excellent agreement with the
interferometer result of 108.73 nm for this surface (.DELTA.
thickness=1.27 nm). Since ellipsometer measurements are considered
correct within thin film technology hierarchy, and both the convex
and flat surfaces are assumed to have identical coating
thicknesses, the b-term method for thin films on curved surfaces
has been additionally verified to have an error of only about 1
nm.
Finally, the previously developed Statistica software program was
applied to tear film lipid spectra, where the input data are
v5=R(.lamda.) meas tear lipid sample.times.R(.lamda.) abs BK7
reference (BK7 abs R calc)/100 and where v1=n.sub.0 air=1,
v2=n.sub.1(.lamda.) lipid, v3=n.sub.2 (.lamda.) aqueous,
v4=measured .lamda., the variable `a` is the fitted lipid film
thickness, and the variable `b` is the final correction term which
moves the measured spectrum up or down on the theoretical R axis
(y-axis) to achieve a match with theory (FIG. 11).
The measured spectra in FIG. 11 were plotted by dividing v5 by the
fitted b-term at each wavelength. FIG. 11 shows lipid film
thicknesses varying from 13.75 nm to 23.03 nm, 53.86 nm, 107.76 nm
and 124.37 nm. It should be noted that the tear spectra in FIG. 11
include cosine-function oscillations from the aqueous layer (the
smaller oscillations). These can be subtracted using a modified
software program. Also, it is clear from the spectrum of the 188.6
nm SiO2 standard (not shown), that spectral data beyond 950-1000 nm
may involve some optical error, perhaps from optical aberration.
The spectrum of the 48.26 nm SiO.sub.2 standard used with the
curved BK7 reference in FIG. 7 also shows some optical error above
950-1000 nm. Thus, a refined software program may delete data
beyond 950 nm. Nonetheless, it is seen that the lipid spectra match
the theoretical spectra very well. Note, these spectra were
acquired over 504 msec, to simultaneously measure the aqueous
layer. It is known that the lipid layer thickness may change over
this time interval. This can cause measured spectra such as the
53.86 nm spectrum to deviate somewhat from theory. In various
embodiments, spectra will be acquired in intervals as short as
20-100 msec to resolve this question. Alternatively, the shape of
the 53.86 nm spectrum may arise from lipid film thickness variation
within the 133 um.times.12.5 um spot. In various embodiments, the
spot size will be reduced to resolve this question.
A modified Statistica software program was created, using a series
of input values, where the input data are v6-v155=R(.lamda.) meas
tear lipid samples and where v1=n.sub.0 air=1, v2=n.sub.1(.lamda.)
lipid, v3=n.sub.2 (.lamda.) aqueous, v4=measured .lamda. and where
v5=R(.lamda.) abs BK7 reference (BK7 abs R calc)/100 and where the
variable a=the fitted lipid film thickness and the variable b=the
final correction term which moves the measured spectrum up or down
on the theoretical R axis (y-axis) to achieve a match with theory.
Here the Euler equation becomes:
v6-v155=(1-((8*v1*v2**2*v3)/((v1**2+v2**2)*(v2**2+v3**2)+4*v1*v2**2*v3+((-
v1**2-v2**2)*(v2**2-v3**2)*(cos(4*3.14159*v2*a*0.98666/v4))))))*b/v5.
Statistica software program code follows for the first several
spectrum calculations (v6 and v7). Here, the starting value for
lipid thickness=65 nm=a-term starting value. The b-term starting
value is set to 0.66. Measured spectra wavelength is edited to
575-950 nm:
TABLE-US-00004 S1.DeleteCases 1, 30 and S1.DeleteCases 730, 994
Option Base 1 Sub Main Dim AO As AnalysisOutput Dim AWB As Workbook
Dim S1 As Spreadsheet Set S1 = ActiveDataSet S1.DeleteCases 1, 30
S1.DeleteCases 730, 994 Dim newanalysis2 As Analysis Set
newanalysis2 = Analysis (scNonlinearEstimation, S1) With
newanalysis2.Dialog .NonlinearMethod =
scNlnUserSpecifiedRegressionLeastSquares End With newanalysis2.Run
With newanalysis2.Dialog .UserFunction = "v6 = ((1-
((8*v1*v2**2*v3)/((v1**2+v2**2)*(v2**2+v3**2)+4*v1*v2**2*v3+
((v1**2-v2**2)*(v2**2-v3**2)*(cos(4*3.14159*v2*a*0.98666/v4)))))))*
b/v5" .CasewiseDeletionOfMD = True End With newanalysis2.Run With
newanalysis2.Dialog .EstimationMethod = scNlnLevenbergMarquardt
.MaxNumberOfIterations = 50 .ConvergenceCriterion = 6 .StartValues
= "65 .66 " End With newanalysis2.Run With newanalysis2.Dialog
.AlphaForLimits = 95 .PLevelForHighlighting = 0.05 End With Set AO
= newanalysis2.RouteOutput(newanalysis2.Dialog.Summary) AO.Visible
= True If AO.HasWorkbook Then Set AWB = AO.Workbook Else Set AWB =
Nothing End If newanalysis2.GoBack With newanalysis2.Dialog
.UserFunction = "v7 = ((1-
((8*v1*v2**2*v3)/((v1**2+v2**2)*(v2**2+v3**2)+4*v1*v2**2*v3+
((v1**2-v2**2)*(v2**2-v3**2)*(cos(4*3.14159*v2*a*0.98666/v4)))))))*
b/v5" .CasewiseDeletionOfMD = True End With newanalysis2.Run With
newanalysis2.Dialog .EstimationMethod = scNlnLevenbergMarquardt
.MaxNumberOfIterations = 50 .ConvergenceCriterion = 6 .StartValues
= "65 .66 " End With newanalysis2.Run With newanalysis2.Dialog
.AlphaForLimits = 95 .PLevelForHighlighting = 0.05 End With Set AO
= newanalysis2.RouteOutput(newanalysis2.Dialog.Summary) AO.Visible
= True If AO.HasWorkbook Then Set AWB = AO.Workbook Else Set AWB =
Nothing End If
The remaining program code follows the above repeating sequence for
additional spectra calculations.
A sample portion of the v1-v6 inputs for a single spectrum (columns
1-6, left to right, where v6=measured % reflectance for tear lipid
spectrum #1) follows in Table 4. Note these columns extend to the
last measured wavelength (1085.11 nm, not shown), and the first 30
rows (shown as input data examples here) and rows where
.kappa..gtoreq.950.6843 nm are deleted by the software.
TABLE-US-00005 TABLE 4 1 1.48173057 1.33783826 559.409653
0.00042340272 36.8916 1 1.48165317 1.33782012 559.929243
0.000423370425 36.1339 1 1.48157604 1.33780203 560.448833
0.000423338215 36.167 1 1.48149916 1.33778399 560.968423
0.00042330609 36.2099 1 1.48142254 1.337766 561.488013
0.000423274049 36.5871 1 1.48134618 1.33774806 562.007604
0.000423242093 36.8416 1 1.48127008 1.33773018 562.527194
0.000423210221 36.2501 1 1.48119423 1.33771234 563.046784
0.000423178432 36.5386 1 1.48111863 1.33769455 563.566375
0.000423146727 35.974 1 1.48104329 1.33767681 564.085965
0.000423115104 35.8217 1 1.48096819 1.33765911 564.605555
0.000423083565 35.6451 1 1.48089335 1.33764147 565.125145
0.000423052107 35.7768 1 1.48081875 1.33762388 565.644736
0.000423020732 35.82 1 1.48074441 1.33760633 566.164326
0.000422989438 35.8625 1 1.48067031 1.33758883 566.683916
0.000422958225 35.8153 1 1.48059646 1.33757138 557.203506
0.000422927093 35.9704 1 1.48052285 1.33755398 557.723096
0.000422896042 35.9033 1 1.48044948 1.33753663 568.242687
0.000422865072 35.8368 1 1.48037636 1.33751932 568.762277
0.000422834181 35.5937 1 1.48030348 1.33750206 569.281867
0.00042280337 35.6052 1 1.48023084 1.33748485 569.801457
0.000422772638 35.6605 1 1.48015843 1.33746768 570.321048
0.000422741985 35.3016 1 1.48008627 1.33745056 570.840638
0.000422711411 35.7279 1 1.48001434 1.33743349 571.360228
0.000422680916 35.6287 1 1.4799428 1.3374165 571.878762
0.00042265056 35.2528 1 1.47987134 1.33739952 572.398352
0.00042262022 34.7073 1 1.47980012 1.33738259 572.917943
0.000422589957 35.3448 1 1.47972913 1.3373657 573.437533
0.000422559772 35.0115 1 1.47965838 1.33734886 573.957123
0.000422529664 35.5492 1 1.47958785 1.33733207 574.476713
0.000422499632 35.069 1 1.4795177 1.33731535 574.995248
0.000422469737 35.1151 1 1.47944763 1.33729865 575.514838
0.000422439857 34.7243
FIG. 12 illustrates the results of using the lipid thickness method
herein to measure tear lipid layer thickness 150.times. over a 25.2
second period. It was found that in a few cases, a 65 nm starting
value for the a-term resulted in a correct, but negative value, for
the thickness. In any case, such results occur infrequently. A test
of the current method with these 150 spectra produced only 5 such
results (3.3%), which is considered acceptable. It is not currently
known why correct values with negative signs are observed. All of
the negative-sign results observed thus far have occurred with
lipid layer thickness values less than 39.45 nm, which are further
away from the 65 nm starting value than many spectra. In any case,
positive-value results are obtained for the aforementioned
negative-value spectra by using a lower starting value for the
a-term. Aqueous layer thickness and blinking were measured
simultaneously according to known methods. Blinks are easily
visualized by the spiking in the aqueous layer thickness at the
same time as the blink. This technological capability to accurately
and quickly measure the tear film lipid layer has not previously
been demonstrated. The results show that the lipid layer averages
61.0 nm and thickens on average about 50 nm very quickly after a
blink, within on average 0.588 seconds. These results are generally
consistent with Korb, et. al, (Korb, D R, et. al. Tear Film Lipid
Layer Thickness as a Function of Blinking. Cornea 13 (4):354-359.
1994), who showed that individuals with a lipid layer thickness of
75-150 nm demonstrated a mean increase in lipid layer thickness of
33 nm following forceful blinking They are also consistent with
Goto, et. al, (Goto, E and Tseng, C G. Differentiation of Lipid
Tear Deficiency Dry Eye by Kinetic Analysis of Tear Interference
Images. Arch Ophthalmol. Vol. 121, February 2003, 173-180.), who
showed that for those with 75 nm lipid films, mean lipid spread
time following a blink was 0.36.+-.0.22 seconds. FIG. 12 shows that
contrary to one conventional theory, lipid layer thickening
following a blink does not precede aqueous layer thickening.
However, this is a single small test of the technology, and not a
rigorous test of tear film spreading theory.
In various embodiments, the disclosed methods may be carried out on
a computing system in communication with an interferometer (e.g. a
wavelength-dependent interferometer). The computing system may
include one or more computer systems in communication with one
another through various wired and wireless communication means
which may include communications through the Internet and/or a
local network (LAN). Each computer system may include an input
device, an output device, a storage medium (including non-transient
computer-readable media), and a processor such as a microprocessor.
Possible input devices include a keyboard, a computer mouse, a
touch screen, and the like. Output devices include a cathode-ray
tube (CRT) computer monitor, a LCD or LED computer monitor, and the
like. Storage media may include various types of memory such as a
hard disk, RAM, flash memory, and other magnetic, optical,
physical, or electronic memory devices. The processor may be any
suitable computer processor for performing calculations and
directing other functions for performing input, output,
calculation, and display of data in the disclosed system.
Implementation of the computing system may include generating a set
of instructions and data that are stored on one or more of the
storage media and operated on by a controller. Thus, one or more
controllers may be programmed to carry out embodiments of the
disclosed invention. The data associated with the system may
include image data, numerical data, or other types of data.
CONCLUSION
Novel mathematical algorithms and software methods have been
independently developed to calculate absolute-reflectance of tear
film lipid layers from measured tear film lipid layer reflectance
using wavelength-dependent optical interferometry. The absolute
reflectance measurements are used for the accurate and quick
determination of lipid layer thickness. These algorithms are
consistent with optical theory, with the exception of a single
b-term, which may be empirically explained by light reflection from
a curved surface or from non-orthogonal placement of a flat surface
or from out-of-focus light reflection.
Thickness errors for the methods herein for thin films on curved
surfaces in perfect focus are only a few nanometers (nm). In
practice, collecting .gtoreq.50 tear lipid measurements, deleting
out-of-focus spectra, and averaging the remaining spectra will keep
lipid thickness errors small.
These methods are suitable for the quantitative evaluation of the
effects of novel dual-function lipid-supplementation tear formulas
on the tear film lipid layer. They are also useful for evaluating
the effects of other eye drops, ophthalmic dry eye drugs and MPS
solutions, and contact lenses on the tear film lipid layer.
Novel features of the present disclosure include the use of the
expanded Euler equation with interferometer-dependent wavelength
selection of wavelength-dependent Sellmaier equation-fitted complex
refractive indices in the software program to calculate tear film
lipid layer thickness, where v6, the measured reflectance variable
R(.lamda.) in an expanded Euler equation, is known and the actual
lipid thickness d becomes the fitted lipid film thickness variable
"a" (e.g., variable reversal in the expanded Euler equation) and
wherein the expanded Euler equation also has a variable b, which is
the final correction term which mathematically adjusts measured
reflectance R (moves the measured spectrum up or down on the
theoretical R axis (y-axis)) to achieve a match with theory:
v6=(1-((8*v1*v2**2*v3)/((v1**2+v2**2)*(v2**2+v3**2)+4*v1*v2**2*v3+((v1**2-
-v2**2)*(v2**2-v3**2)*(cos(4*3.14159*v2*a*0.98666/v4))))))*b/v5
where
v1=n.sub.0 air=1,
v2=n.sub.1(.lamda.) lipid (Sellmeier-form),
v3=n.sub.2 (.lamda.) aqueous,
v4=measured .lamda., and
v5=R(.lamda.) abs BK7 reference (BK7 abs R calc)/100 and
wherein a Levenberg-Marquardt algorithm is used with a novel
software program and a starting value for the a-term of 65 nm and
for the b-term between 0 and 1.30 so that the program achieves the
correct thickness and wherein spectral data without optical
aberration between 575-950 nm are most preferred.
Other novel features include a method wherein a tear film lipid
spectrum and slope is evaluated and a tear lipid layer thickness is
estimated and this thickness estimate is thereafter used as the
starting value for the a-term in the method above.
REFERENCES
The following references are herein incorporated by reference in
their entirety: Scaffidi, R C, Korb, R. Comparison of the Efficacy
of Two Lipid Emulsion Eyedrops in Increasing Tear Film Lipid Layer
Thickness. Eye & Contact Lens: Science & Clinical Practice,
2007; 33(1):38-44. Goto, et al. Computer-Synthesis of an
Interference Color Chart of Human Tear Lipid Layer, by a
Colorimetric Approach. Invest. Ophthalmol. Vis. Sci., 2003;
44:4693-4697. Tiffany, J M. Refractive index of meibomian and other
lipids. Current Eye Research, 5 (11), 1986, 887-889. Stenzel, O.
The Physics of Thin Film Optical Spectra. Editors: G. Ertl, H. Luth
and D. Mills. Springer-Verlag Berlin Heidelberg 2005: 71-98. Schott
technical information document TIE-29 (2005). Korb, D R, et. al.
Tear Film Lipid Layer Thickness as a Function of Blinking Cornea 13
(4):354-359. 1994. Goto, E and Tseng, C G. Differentiation of Lipid
Tear Deficiency Dry Eye by Kinetic Analysis of Tear Interference
Images. Arch Ophthalmol. Vol. 121, February 2003, 173-180.
Various features and advantages of the invention are set forth in
the following claims.
* * * * *
References